Hybrid flow shop scheduling and vehicle routing by considering holding costs

In this paper, a new model for hybrid flow shop scheduling is presented in which after the production is completed, each job is held in the warehouse until it is sent by the vehicle. Jobs are charged according to the storage time in the warehouse. Then they are delivered to customers by means of routing vehicles with limited and equal capacities. The problem’s goal is finding an integrated schedule that minimizes the total costs, including transportation, holding, and tardiness costs. At first, a mixed-integer linear programming (MILP) model is presented for this problem. Due to the fact that the problem is NP-hard, a hybrid metaheuristic algorithm based on PSO algorithm and GA algorithm is suggested to solve the large-size instances. In this algorithm, genetic algorithm operators are used to update the particle swarm positions. The algorithm represents the initial solution by using dispatching rules. Also, some lemma and characteristics of the optimal solution are extracted as the dominance rules and are integrated with the proposed algorithm. Numerical studies with random problems have been performed to evaluate the effectiveness and efficiency of the suggested algorithm. According to the computational results, the algorithm performs well for large-scale instances and can generate relatively good solutions for the sample of investigated problems. On average, PGR performs better than the other three algorithms with an average of 0.883. To significantly evaluate the differences between the algorithms’ solutions, statistical paired sample t-tests have been performed, and the results have been described for the paired algorithms.